Recall that \(\nabla _x c(x,y) = -\log _x(y)\)  [1]}, which is a continuous function of \(y\)  in vicinity of \(y_*\) . Therefore there exists an \({\epsilon }^{\prime }>0\)  so that if \(y\in B(y_*,{\epsilon }^{\prime })\)  we have that \(\left\Vert -\log _x(y)+\log _x(y_*)\right\Vert <\rho \) , where the norm is the Riemannian norm in the tangent space at \(x\) , i.e., \(T_x{\mathcal {M}}\) .
